A059756 Erdős-Woods numbers: the length of an interval of consecutive integers with property that every element has a factor in common with one of the endpoints.

16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96, 100, 106, 112, 116, 118, 120, 124, 130, 134, 142, 144, 146, 154, 160, 162, 186, 190, 196, 204, 210, 216, 218, 220, 222, 232, 238, 246, 248, 250, 256, 260, 262, 268, 276, 280, 286, 288, 292, 296, 298, 300, 302, 306, 310, 316, 320, 324, 326, 328, 330, 336, 340, 342, 346, 356, 366, 372, 378, 382, 394, 396, 400, 404, 406, 408, 414, 416, 424, 426, 428, 430

Offset: 1

Author

Nik Lygeros (webmaster(AT)lygeros.org), Feb 12 2001

Keywords

• nonn

Comments

"Length" means total number of terms including endpoints, minus 1.

Woods was the first to find such numbers, Dowe proved there are infinitely many and Cégielski, Heroult and Richard showed that the set is recursive.

This seems to coincide with prime partitionable numbers in sense of Holsztynski & Strube: n such that there is a partition {P1,P2} of the primes less than n such that for any composition n1+n2=n, there is (p1,p2) in P1 x P2 such that p1|n1 or p2|n2. - M. F. Hasler, Jun 29 2014; there is now a proof for this (see Gribble link), Dec 17 2014

In popular culture: this sequence was involved in the encryption of a message in Episode "eps2.9_pyth0n-pt1.p7z" of the "Mr. Robot" TV series (first aired Sep 14 2016). - Jessica K. Sklar, Jan 30 2019

Named after the Hungarian mathematician Paul Erdős (1913-1996) and the Australian mathematician Alan Robert Woods (1953-2011). - Amiram Eldar, Jun 20 2021

Examples

			a(1) = 16 refers to the interval 2184, 2185, ..., 2200. The end points are 2184 = 2^3 *3 *7 *13 and 2200 = 2^3 *5^2 *11, and each number 2184<=k<=2200 has at least one prime factor in the set {2,3,5,7,11,13}.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 1981, related to Sections B27, B28, B29.
• Konstantin Lakkis, Number Theory [in Greek], Revised edition, 1984.

Links

• Bertram Felgenhauer, Table of n, a(n) for n = 1..10000
• Patrick Cégielski, François Heroult and Denis Richard, On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity, Theor. Comp. Sci., Vol. 303, No. 1 (2003), pp. 53-62.
• David L. Dowe, On the existence of sequences of co-prime pairs of integers, J. Austral. Math. Soc. Ser. A, Vol. 47, No. 1 (1989), pp. 84-89.
• Paul Erdős and John L. Selfridge, Complete prime subsets of consecutive integers, Proceedings of the Manitoba Conference on Numerical Mathematics, 1971, pp. 1-14.
• Bertram Felgenhauer, Some OEIS computations (includes the terms of this sequence up to 10^6).
• James Grime and Brady Haran, Erdős-Woods Numbers, Numberphile video (2024)
• M. F. Hasler and R. J. Mathar, Corrigendum to "Paths and circuits in finite groups", Discr. Math. 22 (1978) 263, arXiv:1510.07997 [math.NT], 2015.
• Christopher Hunt Gribble, Seqfan thread, Dec 05 2014.
• W. Holsztynski and R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272.
• Internet Movie Database, Mr. Robot: Season 2, Episode 11: eps2.9_pyth0n-pt1.p7z
• Nik Lygeros, Erdos-Woods Numbers, contains a list for a(n)<=400000. [Warning: a lot of terms are missing. The smallest missing term is a(169) = 796. - _Jianing Song_, Mar 08 2021]
• William T. Trotter, Jr. and Paul Erdős, When the Cartesian product of directed cycles is Hamiltonian, J. Graph Theory, Vol. 2, No. 2 (1978), pp. 137-142.
• Wikipedia, Erdős-Woods number.
• Alan Robert Woods, Some Problems in Logic and Number Theory, and their Connections, Thesis, University of Manchester, 1981 (reprinted in New Studies in Weak Arithmetics, Sept 2013, CSLI). [Cached copy at the Wayback Machine]

Crossrefs

See A059757 for first terms of corresponding intervals. Cf. A111042, A342310.

Programs

PARI

prime_part(n)=my(P=primes(primepi(n-1)));forstep(x1=2,2^#P-1,2, P1=vecextract(P,x1); P2=setminus(P,P1); for(n1=1,n-1, bittest(n-n1,0) || next; setintersect(P1,factor(n1)[,1]~) || setintersect(P2,factor(n-n1)[,1]~) || next(2)); return([P1,P2])) \\ M. F. Hasler, Jun 29 2014

Extensions

Further terms from Victor S. Miller, Sep 29 2005